Optimal. Leaf size=388 \[ \frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2+105 c^2 d^4\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{210 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}}+\frac {4 d e x \sqrt {a+c x^4} \left (5 c d^2-3 a e^2\right )}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {4 \sqrt [4]{a} d e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (5 c d^2-3 a e^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c} \]
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Rubi [A] time = 0.42, antiderivative size = 386, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1207, 1888, 1198, 220, 1196} \[ \frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (5 \left (5 a^2 e^4-42 a c d^2 e^2+21 c^2 d^4\right )+84 \sqrt {a} \sqrt {c} d e \left (5 c d^2-3 a e^2\right )\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{210 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}}+\frac {e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{21 c^2}+\frac {4 d e x \sqrt {a+c x^4} \left (5 c d^2-3 a e^2\right )}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {4 \sqrt [4]{a} d e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (5 c d^2-3 a e^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 1196
Rule 1198
Rule 1207
Rule 1888
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx &=\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {\int \frac {7 c d^4+28 c d^3 e x^2+e^2 \left (42 c d^2-5 a e^2\right ) x^4+28 c d e^3 x^6}{\sqrt {a+c x^4}} \, dx}{7 c}\\ &=\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {\int \frac {35 c^2 d^4+28 c d e \left (5 c d^2-3 a e^2\right ) x^2+5 c e^2 \left (42 c d^2-5 a e^2\right ) x^4}{\sqrt {a+c x^4}} \, dx}{35 c^2}\\ &=\frac {e^2 \left (42 c d^2-5 a e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {\int \frac {5 c \left (21 c^2 d^4-42 a c d^2 e^2+5 a^2 e^4\right )+84 c^2 d e \left (5 c d^2-3 a e^2\right ) x^2}{\sqrt {a+c x^4}} \, dx}{105 c^3}\\ &=\frac {e^2 \left (42 c d^2-5 a e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}-\frac {\left (4 \sqrt {a} d e \left (5 c d^2-3 a e^2\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{5 c^{3/2}}+\frac {\left (105 c^2 d^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{105 c^2}\\ &=\frac {e^2 \left (42 c d^2-5 a e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {4 d e \left (5 c d^2-3 a e^2\right ) x \sqrt {a+c x^4}}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {4 \sqrt [4]{a} d e \left (5 c d^2-3 a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {\left (105 c^2 d^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{210 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.21, size = 203, normalized size = 0.52 \[ \frac {e x \left (-25 a^2 e^3+28 c d x^2 \sqrt {\frac {c x^4}{a}+1} \left (5 c d^2-3 a e^2\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^4}{a}\right )+2 a c e \left (105 d^2+42 d e x^2-5 e^2 x^4\right )+3 c^2 e x^4 \left (70 d^2+28 d e x^2+5 e^2 x^4\right )\right )+5 x \sqrt {\frac {c x^4}{a}+1} \left (5 a^2 e^4-42 a c d^2 e^2+21 c^2 d^4\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^4}{a}\right )}{105 c^2 \sqrt {a+c x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{4} x^{8} + 4 \, d e^{3} x^{6} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{2} + d^{4}}{\sqrt {c x^{4} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{4}}{\sqrt {c x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 506, normalized size = 1.30 \[ \frac {\sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, d^{4} \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {4 i \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )\right ) \sqrt {a}\, d^{3} e}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+6 \left (-\frac {\sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, a \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, c}+\frac {\sqrt {c \,x^{4}+a}\, x}{3 c}\right ) d^{2} e^{2}+4 \left (\frac {\sqrt {c \,x^{4}+a}\, x^{3}}{5 c}-\frac {3 i \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )\right ) a^{\frac {3}{2}}}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, c^{\frac {3}{2}}}\right ) d \,e^{3}+\left (\frac {\sqrt {c \,x^{4}+a}\, x^{5}}{7 c}+\frac {5 \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, a^{2} \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{21 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, c^{2}}-\frac {5 \sqrt {c \,x^{4}+a}\, a x}{21 c^{2}}\right ) e^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{4}}{\sqrt {c x^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x^2+d\right )}^4}{\sqrt {c\,x^4+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.17, size = 214, normalized size = 0.55 \[ \frac {d^{4} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {d^{3} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 d^{2} e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {d e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {e^{4} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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